Step by step solution to "Set f(x)=x2+ax+b.Prove max{|f(−1)|, |f(0)|, |f(1)|}≥12"

alparcero97oy

Answered question

2022-04-01

Set

f (x) = x^{2} + a x + b

.
Prove

max {| f (- 1) |, | f (0) |, | f (1) |} \geq \frac{1}{2}

Esteban Sloan

Beginner2022-04-02Added 21 answers

Step 1
Let

max {| f (- 1) |, | f (0) |, | f (1) |} = k .

Thus, by your work

2 = (- 2 b) + (1 - a + b) + (1 + a + b) \leq

\leq 2 | b | + | 1 - a + b | + | 1 - a + b | \leq 2 k + k + k = 4 k,

which gives

k \geq \frac{1}{2}

Carter Lin

Beginner2022-04-03Added 13 answers

Step 1
Prove by contradiction. Suppose the maximum is less than

\frac{1}{2}

.
Then

| 1 - a + b | < \frac{1}{2}

and

| 1 + a + b | < \frac{1}{2}

.
Can you use these to show that

| 2 (1 + b) | < 1

?
If you do that you get

1 = | (1 + b) - b | < \frac{1}{2} + \frac{1}{2} = 1

which is a contradiction.

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